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Exotic Aspherical Manifolds a Quick Tour Through Davis's Zoo of Aspherical Manifolds

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Exotic Aspherical Manifolds a Quick Tour Through Davis's Zoo of Aspherical Manifolds Exotic Aspherical Manifolds A quick tour through Davis's zoo of aspherical manifolds Clara Loh Spring 2009 In a series of three talks, we give an introduction to Davis’s construction of peculiar aspherical manifolds and give a quick tour through the resulting zoo of aspherical manifolds. These notes contain a primer of the subject (describing the basic tools and results), a schedule for the three talks, as well as some illustrative material. 1 A quick tour through the quick tour The class of aspherical manifolds plays an important rˆolein various fields of topology. For instance, according to the Borel conjecture, aspherical manifolds should be very rigid topological objects. Two natural perspectives on aspherical manifolds are established by the questions “Which manifolds have a contractible universal covering?” and “Which groups admit a model for the classifying space that happens to be a manifold?”. The condition for a topological space to be both a manifold and aspherical is very restrictive and for a long time basically the only known examples were the ones given by non-positively curved Riemannian manifolds and fibre bundle constructions. The perception of aspherical manifolds changed drastically when Davis in his landmark paper [2] provided the first construction of exotic aspherical manifolds – more specifically, of closed, aspherical manifolds whose universal covering is not homeomorphic to Euclidean space(!). Subsequently, Davis and others refined this technique to construct closed, aspherical manifolds with exotic fundamental groups and exotic geometric properties [3, 4]. 2 1 A quick tour through the quick tour Mirror, mirror on the wall . The key idea underlying Davis’s construction is to glue copies of an aspher- ical space along certain subspaces, called mirrors, according to the com- binatorics of an abstract reflection group – when the input space and the reflection group are chosen appropriately, the glued space is an aspherical manifold; taking quotients then leads to closed, aspherical manifolds. In the following, we describe these steps in more detail: in – Input. We start with a compact, aspherical space X together with a ∂X X closed subspace ∂X, which is triangulated as a flag complex. s t – The associated Coxeter group and mirror structure. From the flag S = = {s, t} ∂X complex ∂X we can read off the presentation of a group W ; namely, 2 2 ∼ W = hs, t | s , t i = D∞ we take the set S of vertices as generators (of order 2) and for every simplex of ∂X, we introduce the relation that the product of the ver- tices of this simplex has order 2. It turns out that (W, S) is an abstract reflection group – more precisely, a right-angled Coxeter group. As mirrors on X we choose the family (X ) of subspaces X X X s s∈S s s t of ∂X that are – in a certain sense – fixed under s ∈ S ⊂ W . More X explicitly, for s ∈ S, the space Xs is just the closed star of s in the barycentric subdivision of ∂X. – The gluing construction. The fundamental step of the whole construc- t s t s tion is gluing copies of X along the mirrors (Xs)s∈S of X via the combinatorics given by the Coxeter group (W, S). The resulting space is denoted by U(W, X), and carries a proper, cocompact W -action. At this point it is crucial that we are dealing with a Coxeter group and that the mirror structure on X is also linked to this Coxeter group. In fact, this guarantees that we can write U(W, X) as an increasing union of aspherical pieces and hence that U(W, X) is aspherical. – Finding a compact quotient. In general, the space U(W, X) will be Γ = h(st)2i ⊂ W non-compact. Because W acts cocompactly on U(W, X), for every subgroup Γ of W of finite index, the quotient U(W, X)/Γ is compact. U(W, X)/Γ ∼ = Furthermore, the obvious retraction U(W, X) −→ X induces a retrac- tion U(W, X)/Γ −→ X. In particular, if (X, ∂X) is a compact aspherical manifold with boundary and if Γ is a torsion-free subgroup of W of finite index, then U(W, X)/Γ is a closed, aspherical manifold that retracts onto X. 1 A quick tour through the quick tour 3 Tailoring aspherical manifolds By adapting certain parameters of the construction described above, we can tailor aspherical manifolds with peculiar properties. Most of these adapted constructions provide examples of the following two types: – Aspherical manifolds with exotic fundamental groups. By thickening finite simplicial models, for every group G of type F we can find a model (X, ∂X) of BG that is a manifold with (triangulated) boundary. Applying the assembly line above to the input (X, ∂X) results in a closed, aspherical manifold that retracts onto X. In particular, strange groups of type F give rise to closed, aspheri- cal manifolds with exotic fundamental groups. For example, one can conclude that there exist closed, aspherical manifolds whose funda- mental group is not residually finite or whose fundamental group has unsolvable word problem. – Aspherical manifolds with exotic geometric properties. Choosing as inputs aspherical spaces (X, ∂X) with exotic geometric properties, also the closed, aspherical manifolds M resulting from the assembly line above will have exotic geometric properties. For instance, starting with a triangulable homology sphere ∂X, we can always find a contractible manifold X whose boundary coincides with ∂X. Then M is a closed, aspherical manifold. If ∂X is not simply connected, then the universal covering (which is U(W, X) in this case) of M is not simply connected at infinity. In particular, if dim M > 2, then M is not covered by Euclidean space. Another interesting choice is to take a compact, aspherical mani- fold (X, ∂X) with triangulated boundary whose Spivak normal fibra- tion does not reduce to a linear vector bundle. Then the Spivak normal fibration of M does not reduce to a linear vector bundle and hence M is an example of a closed, aspherical manifold that is not smoothable. A comprehensive account of the applications emerging from such con- structions is part of Davis’s book [4], covering also various consequences for group cohomology as well as the aspect of curvature. 4 2 Schedule 2 Schedule First session { Reflection groups Introduction and Overview (Clara L¨oh;15 min.). A brief introduction into the subject (cf. Section 1) describing the fundamental questions, tools and results, including an overview of the topics of the talks. Abstract reflection groups (Christian Siegemeyer; 75 min.). This talk provides an introduction to abstract reflection groups (i.e., to Coxeter groups) – starting from the basic terminology for Coxeter groups (Coxeter systems, reflection systems, spherical subsets/cosets, . ), then proceeding to basic algebraic properties of Coxeter groups (in particular, cosets of special subgroups), and ending with the com- plexes associated to Coxeter groups (the nerve, the Davis complex, the Cayley 2-complex). Probably, these concepts are best accompanied with a few elemen- tary running examples (for instance, the infinite dihedral group, sym- metric groups, reflection groups in the model spaces of Riemannian geometry, . ). Literature. [4, Chapter 2.2, Chapter 3 (Theorem 3.3.4), Chapter 4.1 (Theorem 4.1.6), Chapter 7.1, p. 126] · [1, Chapters I and II] Second session { Mirrored spaces Mirrored spaces and the gluing construction (Christian Wegner; 40 min.). Definition of mirrored spaces, the gluing construction, the Davis complex from the perspective of mirrored spaces (including the cell structure and examples). Literature. [4, Chapter 5.1, 5.2, Chapter 7.2–7.4] Algebraic topology of the gluing construction (Wolfgang Steimle; 50 min.). This talk should explain (at least the ideas behind) the computations of the homology, the fundamental group and the funda- mental group at infinity of the gluing construction. In particular, the asphericity criterion for the gluing construction should be presented. Literature. [4, Chapter 8.1, 8.2 (Corollaries 8.2.8, Theorem 8.2.13), Chapter 9 (Theorems 9.1.3, 9.1.5, 9.2.2)] 2 Schedule 5 Third session { Exotic aspherical manifolds The reflection group trick (Thilo K¨ussner;60 min.). Description of the reflection group trick and its applications. A selection of appli- cations should be sketched – e.g., the existence of oriented, closed, connected aspherical manifolds with fundamental groups that are not residually finite, that have unsolvable word problem, etc. A short sketch of the proof that finitely generated Coxeter groups are virtu- ally torsion free should be given. Additionally, the consequences of the reflection group trick from the point of view of the isomorphism conjectures could be discussed. Literature. [4, (Chapter 10.1), Chapter 11.1, 11.2, 11.3?, 11.4?, Chapter 6.12] · [5] Aspherical manifolds not covered by Euclidean space (Clara L¨oh; 30 min.). This talk should sketch Davis’s construction of aspheri- cal manifolds not covered by Euclidean space, including the necessary background on homology spheres (which are a crucial ingredient of this construction). Literature. [4, Chapter 10.3, 10.5] 6 3 Guinea pig – growing a torus familiaris 3 Guinea pig { growing a torus familiaris In the following, we illustrate the basic assembly line for aspherical man- ifolds at a simple example; in particular, this will also present us with an opportunity to get acquainted with the notation involved in the construc- tions. As guinea pig we take the unit square (X, ∂X) := [0, 1]2, ∂([0, 1]2), which is an aspherical manifold with boundary, and equip ∂X with the “obvious” triangulation as a flag complex. s4 s4 s3 s1 s1 s2 The associated Coxeter group. From the flag complex ∂X we read off the right-angled Coxeter group (W, S) given by S := {s1, .
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